# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

**2**

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82 views

### A good notion of “minimal field of definition"

Let $X$ be a variety over a separably closed field $k$.
By definition of variety, there exists a subfield $k_0\subset k$ and a $k_0$-variety $X_0$ such that $$X_0\times_{k_0}k\simeq X$$
There are a ...

**0**

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34 views

### Uniqueness properties and faithfully flat extensions [on hold]

Let $t : A\to B$ be a faithfully flat ring map and suppose $f,g : B\to B$ are two ring endomorphisms.
Assume that there are two ring endomorphisms $f_0, g_0 : A\to A$ such that the following diagram ...

**1**

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22 views

### Counting geometrically irreducible components

If we take a finite field and consider irreducible varieties over it, are there any interesting arithmetical statistics problems associated to the number of geometrically irreducible components?

**21**

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**1**answer

597 views

### Does anybody do $p$-adic Teichmüller theory?

In "Foundations of $p$-adic Teichmüller theory", Mochizuki describes a theory one of whose goals (according to the author) is to generalize Fuchsian uniformization of Riemann surfaces to the $p$-adic ...

**9**

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142 views

### Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves

Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...

**14**

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**3**answers

838 views

### Tower of moduli spaces in Scholze's theory

My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...

**3**

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92 views

### Existence of regular hypersurface sections

Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...

**6**

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**1**answer

127 views

### Endomorphism rings of ordinary elliptic curves

Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}_p$. The discriminant of the Frobenius polynomial is
$\Delta:=t^2-4p.$
So we obtain $4p=t^2-\Delta.$ If $E$ ...

**1**

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192 views

### Moduli spaces of arithmetic varieties with isomorphic $l$-adic cohomology

Given a positive integer $d$, a rational prime $l$ and a number field $K$, is it sensible to consider the moduli stack of $d$-dimensional varieties over $K$ whose $l$-adic cohomology rings are ...

**4**

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159 views

### Does etale homotopy type see the existence of rational points?

Do there exist two smooth projective schemes over $\mathbb{Q}$ that are etale homotopy equivalent and only one of them has a $\mathbb{Q}$-point?

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91 views

### Computing the genus of a plane curve

Let $b(x)=x^4 + 3x^3 + 3x^2 + 2x + 1$, and let $a(x)\in \mathbb Z[x]$ be a separable polynomial. Let $C$ be the plane curve defined by $(y^2+(x+x^2+x^3)a(x))^2-a(x)^2b(x)=0$. I would need to show that ...

**6**

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**1**answer

229 views

### Rigid versus log-rigid cohomology for semistable varieties

If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...

**1**

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82 views

### Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...

**3**

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108 views

### Extension of scalars and power series rings

Let $R$ be a ring and $R[\![x]\!]$ the formal power series ring over $R$.
Suppose $R\to R’$ is either:
a finite étale ring map
a localization at the multiplicative set generated by one element $f\in ...

**5**

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117 views

### Hasse-Weil zeta function of smooth projective toric varieties

Let $X$ be a smooth projective toric variety over a number field $K$ (assume the tori is split). As $X$ is rational, maybe the related Hasse-Weil zeta function can be well-understand, so how much do ...